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Unformatted text preview: STA 4321/5325 — Spring 2010
Quiz 8 — April 9 Name: my There are ﬁve problems in this quiz. Each problem has exactly one correct answer. Problem 1 Let f (as, y) denote the joint probability density function of two continuous random
variables X and Y, and f X(m) denote the marginal probability density function of X. Then it is
always true that (a) fx($) = If; H1737de
(b) fx(w) = Li” f(m, y)dy
(C) fx 2? (d) fx(w)=f:’;f<x,y)dy ' Problem 2 Let X and Y be two discrete random variables with the following joint probability
mass function: Y
Way) 0 1 2 gix‘x) 0 1/9 2/9 1/9 V777,“
X 1 2/9 2/9 0 577 2 1/90 0 E
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ll Problem 3 Let X and Y be two continuous random variables with joint probability density
function f (ac, y), and marginal density functions fX and fy respectively. Then X and Y are
said to be independent if f (2:, y) = fX(£L’)fy(y) for every :3 e R, g E R. This statement is (b) False Problem 4 Let X and Y be continuous random variables taking positive values, with joint
probability density function given by f (m, y) = e’(‘”+y) for every .1: > 0, y > 0. It can be derived
that the marginal probability density function of Y is given by fy(y) = e‘y for y > 0. Then, the
conditional probability density function of X given Y = 7 is given by % C )‘3 I y: C (a) ley:7(m) = 67 for every x > 0 ﬁx] _: ’f 7}—
‘lgr L 7)
7 C (b) fXY=7($) 2 84$”) for every .7: > 0 z (c) ley:7(:r) = 6—9“ for every x > 0)
IX» (d) fXY=7(CI7) = e_(y+7) for every (E > 0 : 6 Problem 5 Let X and Y be two independent Bernoulli random variables. Let P(X = 0) : and P(Y = 0) 2 Then,
ﬁxed=1 r) ﬁx“): Z WIH ll ti)th “Nib (a) P(X =0,Y= 1) ~
(C) p(X=0’Y:1) ‘ {7(120}:1—c7 VLYfJ): (d) P(X = O,Y = 1) ©l‘ «:10. CLE—
3*3
2;. ...
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 Fall '08
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 Probability

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